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Formulae_Mech_page2

# Formulae_Mech_page2 - Oscillation motion 1 f = T = 2 T 2 2...

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Oscillation motion f = 1 T , ω = 2 π T S H M: a = d 2 x dt 2 = - ω 2 x , α = d 2 θ dt 2 = - ω 2 θ x = x max cos( ω t + δ ), x max = A v = - v max sin( ω t + δ ), v max = ω A a = - a max cos( ω t + δ ) = - ω 2 x , a max = ω 2 A E = K + U = K max = 1 2 m ( ω A ) 2 = U max = 1 2 k A 2 Spring: m a = - k x Simple pendulum: m a θ = m α ‘ = - m g sin θ Physical pendulum: τ = I α = - m g d sin θ Torsion pendulum: τ = I α = - κ θ Gravity ~ F 21 = - G m 1 m 2 r 2 12 ˆ r 12 , for r R , g ( r ) = G M r 2 G = 6 . 67259 × 10 - 11 N m 2 /kg 2 R earth = 6370 km, M earth = 5 . 98 × 10 24 kg Circular orbit: a c = v 2 r = ω 2 r = 2 π T · 2 r = g ( r ) U = - G m M r , E = U + K = - G m M 2 r F = - d U dr = - m G M r 2 = - m v 2 r Kepler’s Laws of planetary motion: i ) elliptical orbit, r = r 0 1 - ² cos θ r 1 = r 0 1+ ² , r 2 = r 0 1 - ² ii ) L = r m Δ r Δ t -→ Δ A Δ t = 1 2 r Δ r Δ t = L 2 m = const. iii ) G M a 2 = 2 π a T · 2 1 a , a = r 1 + r 2 2 , T 2 = 4 π 2 G M · r 3 Escape kinetic energy: E = K + U ( R ) = 0 Fluid mechanics Pascal: P = F 1 A 1 = F 2 A 2 , 1 atm = 1 . 013 × 10 5 N/m 2 Archimedes: B = M g , Pascal=N/m 2 P = P atm + ρ g h , with P = F A and ρ = m V F = R P dA -→
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